# Lecture 15 Estimating a Population Proportion

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```					         Sections 7.1 and 7.2

This chapter presents the beginning
of inferential statistics.
The two major applications of inferential
statistics

Estimate a population parameter: proportion, mean
Test some claim (or hypothesis) about a population.
Point estimate: a single number
Interval estimate: interval of numbers.
Confidence Interval

Why?: point estimate is not reliable under
re-sampling.
A confidence interval (CI): an interval of
values used to estimate the true population
parameter.
Point Estimate

p=          population proportion

ˆ= n
p  x         sample proportion
of x successes in a sample of size n.
(pronounced     Unbiased estimate (best estimate)
‘p-hat’)

ˆ       ˆ
q = 1 - p = sample proportion
of failures in a sample size of n
Example: Photo-Cop Survey Responses
829 adult Minnesotans were surveyed, and 51% of them are
opposed to the use of the photo-cop for issuing traffic
tickets. Using these survey results, find the best estimate of
the proportion of all adult Minnesotans opposed to photo-
cop use.

Best point estimate=sample proportion=51%.
Confidence Level

α: between 0 and 1
A confidence level: 1 - α or 100(1- α)%. E.g. 95%.
This is the proportion of times that the confidence
interval actually does contain the population parameter,
assuming that the estimation process is repeated a
large number of times.
Other names: degree of confidence or the
confidence coefficient.
The Critical Value
(z-score)
Given α
Finding zα/2 for 100(1- α)% Confidence Level

α =5%

α/2 = 2.5% = .025
Sampling Distribution of ^
p
The sampling distribution of sample proportion can be
approximated by a normal distribution if np≥15 and
nq ≥15 : phat is approximately N(p, pq/n), q=1-p.
p             p− p
ˆ
z=
ˆˆ
pq
n

p
^
p
^
Margin of Error of p
the maximum likely (with probability 1 – α)
difference between the observed proportion
^
p and the true population proportion p.

E = zα / 2
pˆ
ˆq
n

^
Standard Error of p
=se
Finding the 95% Confidence Interval
for a Population Proportion
A 95% confidence interval for a population
proportion p is:

ˆ     ˆ
p(1 - p)
p ± 1.96(se), with se =
ˆ
n
100(1-α)% confidence interval for p is

p(1 − p)
ˆ     ˆ
p ± zα / 2 ( se)
ˆ                  with   se =
n
Example: Would You Pay Higher
Prices to Protect the Environment?
In 2000, the GSS asked: “Are you willing to
pay much higher prices in order to protect
the environment?”
Of n = 1154 respondents, 518 were willing to
do so
Find and interpret a 95% confidence interval
for the population proportion of adult
Americans willing to do so at the time of the
survey
Example: Would You Pay Higher
Prices to Protect the Environment?

518
p=
ˆ         = 0.45
1154
(0.45)(0.55)
se =                = 0.015
1154
E = 1.96(se) = 1.96(0.015) = 0.03
p ± E = 0.45 ± 0.03 = (0.42, 0.48)
ˆ
What is the Error Probability for the
Confidence Interval Method?
Summary: Effects of Confidence Level
and Sample Size on Margin of Error

The margin of error for a confidence interval:

Increases as the confidence level increases
Decreases as the sample size increases
Determining Sample Size
Recall :
E=    zα / 2       ˆˆ
pq
n

(solve for n by algebra)

n=    zα / 2 p q
ˆˆ
2

E2
Sample Size for Estimating
Proportion p

ˆ
When an estimate p of p is known:

n=    ( zα / 2   )2
ˆˆ
pq
E2
When no estimate of p is known:

( zα / 2)2 0.25
n=           E2
Example: Suppose a sociologist wants to determine       the
current percentage of U.S. households using e-mail. How many
households must be surveyed in order to be 95% confident that the
sample percentage is in error by no more than four percentage
points?

a) Use this result from an earlier study: In 1997, 16.9% of U.S.
households used e-mail (based on data from The World Almanac
and Book of Facts).
b) Assume that we have no prior information suggesting a possible
value of p.
a) Use this result from an earlier study: In 1997, 16.9% of U.S.
households used e-mail (based on data from The World Almanac
and Book of Facts).

n = [za/2 ]2ˆ ˆq
p                      To be 95% confident that our
E2                            sample percentage is within
four percentage points of the
= [1.96]2 (0.169)(0.831)           true percentage for all
0.042                      households, we should
randomly select and survey
= 337.194                          338 households.
= 338 households
b) Assume that we have no prior information suggesting a possible
value of p.

n = [za/2 ]2 • 0.25
E2                      With no prior information,
= (1.96)2 (0.25)             we need a larger sample to
0.042                   achieve the same results
with 95% confidence and an
= 600.25                     error of no more than 4%.
= 601 households
Finding the Point Estimate
and E from a
Confidence Interval
Point estimate of p:
ˆ
p = (upper confidence limit) + (lower confidence limit)
2

Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2

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